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Model-scale ice — Part A: Experiments
Cold Regions Science and Technology 94 (2013) 74–81
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Model-scale ice — Part A: Experiments Rüdiger von Bock und Polach a,b,⁎, Sören Ehlers b, Pentti Kujala a a b
Aalto University, School of Science and Technology, Department of Applied Mechanics, Marine Technology Group, P.O. Box 15300, FI-00076 Aalto, Finland Norwegian University of Science and Technology, Faculty of Engineering Science and Technology, Department of Marine Technology, NO-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 11 October 2012 Accepted 3 July 2013 Keywords: Model-scale ice Experiments Strength measurements Thin plate theory Poisson's ratio Elastic strain-modulus
a b s t r a c t This paper is presenting novel model-scale ice property measurements for grain size, elastic strain-modulus, compressive and tensile specimen tests. The testing and analyzing procedure is targeted to define the basic material behavior accurately to understand the material behavior for the future development of a numerical material model. Additionally, the model-scale ice thickness and the bending strength (following ITTC) are determined to classify the ice properties. The experiments consist of systematic in-situ tests to identify the model-scale ice properties in a format suitable for numerical simulations. The elastic strain-modulus is determined on the intact level ice sheet based on the load displacement relationship of the infinite plate deflection. All specimens are cut with a template to minimize dimensional variations. The specimens are loaded with a linear drive at constant speed while displacement and force are recorded. The resulting load–displacement curves indicate good repeatability. The experiments are conducted over a time of 4 h–5 h in the keeping phase, where the cooling system is adjusted to maintain the mechanical ice properties, and the obtained results do not show a dependency on the time of testing. A linear-elastic finite element model is used to reproduce the plate bending measurements for the elastic strain-modulus determination. Therewith, it is found that the actual elastic strain-modulus is 27% larger than in plain stress theory due to stresses in thickness direction. Additionally, the approximate yield strength of the model-scale ice is investigated and is found to be significantly lower than the determined maximum stresses in compression, tension and bending. Consequently, this paper contributes to a deeper understanding of the mechanics of model-scale ice, and a procedure is shown how the mechanical parameters can be determined by systematic experiments and analyses. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Offshore operations and marine-borne transport systems are reaching more and more towards northern territories to exploit fossil resources and to ship goods through shorter, but ice infested, routes. The navigation in such areas requires the ships to be ice-capable. The Finnish-Swedish Ice-Class Rules TraFi (2011) have set the standard for the lower Polar ice/Baltic ice classes to which most classification societies refer. Those standards state the possibility to prove the ice capability of a ship with ice model tests. Additionally, the performance of most ice going structures and ships is tested in modelscale ice before concluding the design. Hence, the model-scale ice properties are of high significance for the design of marine structures, and a deeper understanding of model-scale ice as a material is important. In relation to full scale, most ice property measurements in model-scale ice can be conducted at low cost. Therefore, the model-scale ice offers the chance to investigate failure ⁎ Corresponding author at: Aalto University, School of Science and Technology, Department of Applied Mechanics, Marine Technology Group, P.O. Box 15300, FI-00076 Aalto, Finland. Tel.: +358 504 059 030. E-mail address: ruediger.vonbock@aalto.fi (R. von Bock und Polach). 0165-232X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coldregions.2013.07.001
processes and material behaviors better than in natural ice. This emphasizes conducting systematic ice property measurements to develop a numerical material based on the knowledge gained from the experiments. The numerical model of the model-scale ice targets future comprehension of marine design processes and simulations. Model-scale ice is grown in a controlled environment and the generated ice has a high potential of repeatability. The water from which the model-scale ice is created contains additives, such as ethanol at the Aalto University Ice Tank that makes the model-scale ice Froudescalable. The ice strength is affected by the properties of the grain boundaries. Elvin and Sunder (1996) described grain boundaries as the weakest link in polycrystalline ice due to their high entropy and disorder. The nature of the grain boundaries depends on the water dopant and the amount dissolved in water, which is reflected in Timco (1980). Therein, an extensive series of mechanical ice property measurements has been presented to explore new dopants to enhance the scalability of the model-scale ice properties. So far, model-scale ice property measurements are not reported in a systematic fashion suitable for a numerical implementation and are in general very scarce. Similarly, the results presented by Timco (1980) are not suitable for the implementation into numerical simulations for in-situ experiments, because the constitution of the ex-situ samples might have changed compared
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2 0
Temperature [C]
to in-situ conditions due to fluid drains (see Li and Riska, 1996; Timco, 1981). Furthermore, it was stated in Timco (1980) that the tensile strength could not be measured directly due to the weak constitution of the model-scale ice. Additionally, Timco (1980) used an empirical concentration factor to calculate the tensile strength, the origin of which might add an uncertainty to the results. Unfortunately, the tensile strength of ice and scaled ice (model-scale ice) is one of the least explored properties due to the difficulty involved in the measurements (see Timco and Weeks, 2010). Furthermore, the experiments of Timco (1980) are missing information on the entire material response up to specimen failure, which is essential for modeling the material behavior. In conclusion, earlier model-scale ice experiments are not suitable for in-situ simulations due to the lack of systematic measurements and the absence of a numerical implementation focus. The controlled model-scale ice-generation process allows adjusting target properties, but nevertheless (see Timco, 1980) the model testing results are subjected to some scatter. Earlier non-scaled ice experiments by Kim and Sunder (1997) found that both the obtained failure strength and the grain size distribution follow a log-normal probability distribution and established a connection between grain structure and failure. This indicates that the grain size distribution is dominating the strength compared to other micro-structural effects. This paper presents systematic ice property measurements and their analysis. The main objective of this paper is to gain a basic understanding of mechanical material behavior to provide sufficient information to develop a numerical material model in a later stage. The later numerical model aims to simulate ice–structure interactions and therefore all experiments are conducted in-situ i.e. the same environment where model-scale experiments are conducted. Model-scale ice is a unique and temperature (time) sensitive material, and constitutional changes between the tested samples are to be prevented. This naturally limits the number of samples in the analysis. Furthermore, all test specimens are of the same size to exclude variations due to geometrical effects. Finally, the grain size measurements are statistically analyzed to increase knowledge of the micro-structure and to investigate their link to the strength measurements in future research.
75
2
1
4
3
−2 −4 −6 −8 −10 −12 −14 0
5
10
15
20
25
30
Time [h] Fig. 1. Temperature time-history, with the phases of 1: spraying (adjustment of the thickness), 2: consolidation (adjustment of the strength), 3: tempering, and 4: keeping.
ice sheets), while the densities of the model-scale ice (911 kg/m3) and the basin water (989 kg/m3) were determined a priori. The density is determined from the buoyancy related reaction force of a submerged ice piece. 3.1. Grain size measurements The grain size measurements are conducted with post-processed photographs visualizing single grains with well-defined grain edges. The photographs are taken on a portable photographer-light-table, where a thin ice sample is placed between two transparent polar filter foils. As the material is very weak and unstable, thin sections are
a
35 30 25
2. The model-scale ice
20
x [m]
The model-scale ice is produced in the ice tank of the Aalto University, which is a square 40 m × 40 m basin with large cooling units over the entire basin widths on two opposite sides controlling the temperature. The model-scale ice is fine grained and is produced by the spraying method. The resulting model ice has fairly homogeneous structure of several crystal layers. More information is found in Jalonen and Ilves (1990), Li and Riska (1996) and Li and Riska (2002). Fig. 1 shows the cooling history and the different phases of cooling. The ice model tests are conducted in the keeping phase (phase 4, see Fig. 1), where sufficient cold air is added to keep the mechanical model-scale ice properties constant. Even though the ice properties are rather constant over the entire ice sheet, it is uncertain as to what extent the temperaturedistribution is a function of the tank location, i.e. the distance to the cooling units. Therefore, the experiments are only conducted in one particular section of the ice sheet, as seen in Fig. 2, where the tripod is covering a diameter of 4 m for size comparison.
15 10 5 0 −5 35
b
30
25
20
15
10
5
0
−5
y [m]
3. Experiments The measured ice properties in the ice sheet are the elastic strain-modulus, E, the bending strength, σb, the tensile strength, σt, and the compressive strength, σc. The latter two strength definitions refer to force divided by cross-sectional area. It is ambiguous whether the strength definitions are correct, but they refer to the state of the art in ice model testing tanks. The grain size is determined in later measurements in an ice sheet of 25 mm (the same thickness as the tested
Fig. 2. Location of experiments in the Aalto ice tank as (a) plan view and (b) testing site.
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Fig. 5. Template for specimen shapes for tension (1), compression (2) and bending (3). The loading direction of each specimen is indicated by the corresponding symbol.
to convert the pixels to millimeters. The length measurement in one direction suffices because the grains are fairly circular in this 2D view. Furthermore, it is assumed that this sample preparation (Fig. 3) is creating randomly orientated crystals, and thus the samples are assumed to represent the grain size in general.
3.2. Elastic strain-modulus
Fig. 3. Photographs of the three used grain size samples (a), (b) and (c).
impractical. The thin slice collapses during the cutting off and therefore the structure of the cross section cannot be analyzed. It is assumed that the collapsing of the slice leads to a random orientation of the grains on the polar filter foil. Subsequently, the polar filter foils are pressed slightly together by hand, without damaging the grains, so that the ice distributes as one spread layer of grains between the two foils. The foils are rotated relative to each other until the highest contrast with the ice is established, so that the single ice grains are clearly visible (see Fig. 3). The colors of the photographs are inverted to raise the clarity of the grain edges (see Fig. 3) and the grain size measurements are performed on three different photographs with different orientations with 20 sample measurements each. A millimeter scale is placed on each photograph for reference
Fig. 4. Measurement setup with the model-scale ice (1), the insulation plate (2), the steel mass (3), the emitted laser beam (4) and the laser (5).
The elastic strain-modulus of the ice sheet is determined from the deflection of the intact ice sheet, which is assumed as an infinite plate on an elastic foundation (following ITTC, 2002). The non-destructive tests are carried out prior to the destructive tensile, compressive and bending experiments. The ice sheet deflection is measured at four spots within the area of the later strength measurements. The ice sheet is loaded with three different certified mass levels of 26.9 g, 55.9 g and 109 g. Each measurement set includes several loading and unloading cycles of one mass level. The laser (5) in Fig. 4 is attached to a light-wooden tripod with a span of 4 m. The laser emits a beam (4) to measure the displacement difference between the loaded and unloaded condition on a coated Divinycell PVC foam plate, which is placed on the ice surface. The plate (2) prevents heat transfer between the steel mass (3) and the model-scale ice (1). However, it can be observed that despite cooling all equipment down to low temperature, a thin wet layer is generated between the equipment contact points and the ice sheet. This melting process leads to a continuous
Fig. 6. Specimen cutting process with specimens for tensile (1), compressive (2) and bending (3) experiments. See Fig. 5 for the related template shapes.
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Table 1 Measured elastic strain-modulus, E, for the ice sheet following ITTC (2002). Location
Mass [g]
Mean E [MPa]
1 2 3 4
26.9 55.9 109.0 109.0
108.2 108.1 Creep Creep
the ice thickness of 25 mm. The dimensions of the cantilever beam follow the ITTC guidelines (see ITTC, 2002). 4. Analysis of the experiments 4.1. Grain size Fig. 7. Test setup of the apparatus and the specimens in-situ with impact cylinder (1), impact plate (2), plunger (3), load-cell (4), linear drive (5) and rotation axis (6).
caving-in (0.1 mm) of the tripod into the ice, which, however, converges to a steady state. Therefore, both the load level and the zero level run parallel in a non-linear progression.
The analysis of the grain size dimensions is done by measuring 20 grains of each sample-photograph (a, b and c, see Fig. 3) for the statistical analysis. The grains selected have clearly visible edges and are best suited to size analysis. The statistical analyses of the 60 independent grain diameter measurements follow a log-normal distribution, with the highest probability density at 0.68 mm (see Fig. 8) and arithmetic mean of 0.72 mm (standard deviation of 0.13 mm).
3.3. Ice strength measurements 4.2. Elastic strain-modulus and ice sheet deflection The experiments are conducted in-situ by cutting specimens with a template in the ice sheet (as can be seen in Fig. 5). The numbers denote specimens for 1 — tension, 2 — compression, and 3 — bending (following ITTC, 2002). The dimensions of the specimens in Fig. 5 are related to the ice thickness, h, as follows: • Tensile specimen outer ring diameter = 3 h, inner ring diameter = h, and thinnest width of dumbbell-part = h • Compressive specimen width = h and length = h • Cantilever beam width = 2.25 h and length = 6.5 h. The specimens are cut with a guided drill to ensure the specimens are cut as vertical projections of the template. After the outlines of the specimens are cut, the surrounding ice is carefully removed by hand (see Fig. 6). Once the specimens are ready the testing device is employed. The testing device is a linear drive (see Fig. 7) equipped with a rigid impact cylinder (1) to apply the tensile loading, a rigid impact plate (2) to apply the compressive loads, a load-cell (4) and a displacement transducer (5). Further, orthogonal to the plate (2) is a plunger (3) for beam bending tests. In the bending tests the apparatus is rotated around axis 6 (Fig. 7) by 90° to load the cantilever beam vertically. The dimensions of all specimens are a function of the ice thickness. The width of the rated breaking point of the tensile specimen and the width of the compressive specimen were equal to 0.995 0.99
Table 1 compiles the results of the elastic strain-modulus measurements, which are analyzed based on the theory of the infinite plate on an elastic foundation. Table 1 shows a good agreement between measurement 1 and 2. As mentioned earlier, the displacement signal is subjected to some drift due to the carving-in of the tripod into the model-scale ice surface. Fig. 9 shows the drift of the displacement signal and the interpolation curves for the loaded sections and the zero levels. As it is known that the signal drift (see also Li and Riska, 1996) would converge to a certain value after the process of caving-in is terminated, a cubic function is chosen instead of a linear interpolation. The time history sections under load should ideally be a curve with a constant offset to the unloaded curve sections. Significant deviations from this offset indicate creep, as reflected in Fig. 10, where the load of 109 g indicates exceedance of the linear-elastic regime. The numerical model is built in LS-Dyna, where the ice sheet is modeled as a homogeneous, isotropic and linear-elastic material, with the material properties elastic strain-modulus, E, and Poisson's ratio, v. The elastic foundation is modeled with load displacement nodes where a displacement of the nodes causes a reaction force equivalent to the buoyancy restoring force. The modeled ice sheet measures 4 m × 4 m with all degrees of freedom constrained at the edges and 25 mm sized elements. The load on the ice sheet is applied
0.1
Grain size measurements Log−normal (N = 60, p = 0.0026)
Displacement [mm]
0.95
Probability
0.9 0.75 0.5 0.25 0.1 0.05 0.01
Measurement Zero−level Load−level
0.05 0 −0.05 −0.1
0.005 0.001 0.5
0.6
0.7
0.8
Grain size [mm] Fig. 8. Measured grain sizes with log-normal probability plot.
0.9
1
0
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20
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40
50
60
70
Time [s] Fig. 9. Displacement measurement of infinite plate bending with 26.9 g loading.
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Table 2 The stresses for the solid model at the maximum displacement at 55.9 g loading.
Displacement [mm]
0.15 0.1 0.05
Effective stress [Pa]
σx [Pa]
σy [Pa]
σz [Pa]
τxy [Pa]
τzy [Pa]
148
449
32
32
57
13.8
172
0 −0.05 −0.1 −0.15
Start of creeping
−0.2 −0.25 0
50
100
150
200
250
300
350
400
Time [s] Fig. 10. Displacement measurement of infinite plate bending with 109 g loading.
as a point load (see also ITTC, 2002). The ice sheet deflection is successfully modeled with shell elements complying with the analytical formulation for a thin plate on an elastic foundation stated in ITTC (2002). However, the remodeling of the plate deflection with fully integrated solids results in a higher deflection by 12.3% (see Fig. 11). Compliance between the displacement of the solid FE-model and the measurements is achieved by increasing the elastic moduli to 25%–27% (145 MPa and 148 MPa). Table 2 compiles the stresses for the 55.9 g mass unit and states that the stresses in throughthickness direction σz and the shear stresses τzy are of significant magnitudes, which are zero for the thin plate theory following plane-stress theory. The ice shows significant inelastic behavior at 109 g loading (see Fig. 10). The maximum effective elastic stress (von Mises) for elastic response is found for the 55.9 g mass unit at 449 Pa (see Table 2). The elastic limit of the material is of high significance in a numerical model. In the linear-elastic domain, the Poisson's ratio is the single parameter affected with uncertainty. Table 3 reflects the sensitivity of the elastic strain-modulus and the maximum effective stress (von Mises) in dependence on the Poisson's ratio at the same maximum displacement.
4.3. Strength measurements The average (downward) bending strength following ITTC (2002), without accounting for buoyancy effects, is found to be 59 kPa. Fig. 12 reflects the directly measured force displacement curve of the compression measurements. Figs. 13 and 14 show the force divided by the cross-sectional area of the specimen at the location of failure. The Figs. 12–14 show that the slopes of all specimens are rather similar. However, the point of failure is different, which is a phenomenon 0.01 0
Displacement [mm]
Elastic strain-modulus [MPa]
Solid model Shell model
−0.01 −0.02 −0.03 −0.04
that is also known from other materials such as concrete and even steel. The compressive and tensile failure processes are analyzed and interpreted in context with the visual observations, i.e. video recordings, and the recorded signals. In compression, the ice appears to undergo a linear deformation until failure. In some cases the slope of the curve reduces prior to failure, which indicates additional softening at a certain stress level. The videos revealed that the failure initiates in the center of the specimen width, where the specimen experiences high confinement. Therefrom, the failure propagates towards the high energetic sides (free surface energy). The crack initiation requires a nucleation base, which may be impurities or small defects in combination with high local confinement. Furthermore, Table 4 states the origin of the failure crack progression. Table 4 does not indicate a correlation between the post-compressive crack pattern and failure load level. The tensile failure occurs at the predetermined breaking point, where the specimen has the thinnest cross-section, as seen in Fig. 16. The crack occurs perpendicular to the direction of load application. Five of the specimens have a crack perpendicular to the load direction and one each with an angle of 74° and 68° to the direction of the load application. As for the compressive tests, no correlation between load level and breaking pattern is indicated. The maximum stresses in the tensile and compressive specimens are determined by assuming that the stresses are evenly distributed over the cross-section. Possible material inhomogeneities are neglected, which complies with current approaches in engineering.
5. Discussion The density of the model-scale ice is measured as 911 kg/m3. The model-scale ice is considered a compound of ice crystals, water voids and air voids. The density of frozen water (ice) is 917 kg/m3, the density of the tank water is 989 kg/m3 (measured) and the density of air is 1.3 kg/m3. Based on this, the volume fractions of the different compounds are determined as 4.5% water inclusions and 1% air inclusions, which is in line with the description of Li and Riska (2002). The grain size distribution is obtained from randomly selected samples of each test. Therefore, the selected grains reflect the global state of the model-scale ice texture. The log-normal distribution is found to fit best to the grain sizes. The grain size distribution of the model-scale ice coincides with the finding of Kim and Sunder (1997) for fresh-water ice, yet it is not clear whether this is coincidence or if there is a direct link. The elastic strain-modulus is determined by the deflection of an elastic plate on an elastic foundation. It is assumed that elastic strain-modulus is constant and isotropic over the thickness. The temperature gradient between top and bottom differs at maximum by 0.05 °C. Therefore, it is assumed that, as in nature, a temperature related variation over the ice thickness of the elastic strain-modulus
−0.05 Table 3 Elastic strain-modulus and effective stress in dependence on the Poisson's ratio.
−0.06 −0.07 0
500
1000
1500
Distance [mm] Fig. 11. The bending lines of plates on an elastic foundation modeled with shells and solids.
Poisson's ratio
Elastic strain-modulus [MPa]
Effective stress [Pa]
0.3 0.2 0.4
148 134 158
449 450.1 448
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40 30 20 10 0
T31
60
T32 T33
50
Force / Area [kPa]
Force [N]
50
−10
70
C32 C33 C37 C38 C39
60
79
T35
40
T36 T38
30
T39
20 10 0
−10 0
1
2
3
4
5
6
7
8
9
10
−20
4
5
6
Displacement [mm]
7
8
9
10
Displacement [mm]
Fig. 12. Displacement-force progression of the tested specimens for which the failure pattern is displayed in Fig. 15.
Fig. 14. Displacement-force per contact area for the tensile measurements.
(see Kerr and Palmer, 1972) or of other mechanical parameters is negligible. The measured displacement signal from the elastic strain-modulus measurements is subjected to drift. The drift is related to a caving-in of the tripod feet into the ice, which converges to a steady state over time. Therefore, the zero and load level are interpolated with a cubic function. The Poisson's ratio, v, is the only parameter subjected to uncertainty, as it is very difficult to determine for model ice due to the weak constitution. Li and Riska (1996) and Timco (1979) (Aalto ice tank) used v = 0.3 for model ice, which is in line with similar values used for full scale scenarios by Gammon et al. (1983) and Svec and Frederking (1981) or more advance mathematical models such as Derradji-Aouat (1992). This value is also in line with the theory of Froude-scaling. Fig. 10 shows that the model-scale ice is subjected to creep/ permanent deformation for a mass of 109 g. The load level of 56 g is the highest test load for which the model-scale ice responded elastically. The corresponding stress state is considered to represent the lower confident yield limit of the model-scale ice. The remaining uncertainty in the Poisson's ratio affects the yield strength (Table 3) negligibly. The finite element analyses with solids and shells deliver a difference in the elastic strain-modulus by 26% for the same displacement. Fig. 11 reflects the significant difference in the bending lines of the solid and the shell model for the same elastic strain-modulus and the same load. Furthermore, the shell model is capable of remodeling the infinite plate on an elastic foundation exactly, following ITTC (2002). The difference in displacement is related to the relatively large stresses (shear and normal) in thickness direction, which are
not regarded in the thin plate theory following the plain stress approach. It must be acknowledged that the plate deflection experiment is a quasi-static test, whereas the compressive, tensile or bending tests are dynamic tests with a loading rate. However, the static test is considered sufficient, which is common practice in all model test basins. The determined yield strength (0. kPa) lies about two orders of magnitude below the value ranges of bending strength, compressive strength or tensile strength. This indicates that the linear elastic range is small and most of the deformation process takes place in a damage driven domain. Herein it is assumed that the yield surface is isotropic. Fig. 17 shows that the linear elastic simulation based on the measured elastic strain-modulus cannot represent the measured load displacement curve. This is in line with the finding of a relatively low yield strength, which indicates that the deformation process must take place in a domain other than the linear elastic one. The strong deviation of the linear elastic force displacement curve from the measured load displacement curve indicates that the occurring effects are not viscoelastic. The progressions in Fig. 17 suggest that after yielding the stress–strain behavior is governed by a lower tangent or softening modulus. Earlier research of Schwarz (1985) and Timco (1980) indicated the elastic-strain modulus as key parameter to be scaled in model-scale ice tests. It is considered to drive the deformation behavior. This complies with the idea of Froude-scaling and the understanding of sea-ice behavior (see Timco and Weeks, 2010), which is considered as mainly elastic. However, it must be acknowledged that research as presented in this paper has not been presented yet for any model-scale ice. In this context references of yield or creep strength of model-scale ice are not available. The constitution of model ice is slushier and differs significantly from sea ice, which is dominated by elastic and viscoelastic effects. This in turn challenges the postulation of Schwarz (1985) that the ratio of elastic strain-modulus and bending strength can be used as quality criterion for the scalability of the model-scale ice of Aalto University or model-scale ice in general. All experiments (tensile, compressive and bending) are conducted at a testing speed of 6 mm/s, as Wegner (2011) indicated that between 6 mm/s and 17 mm/s the
120
Force / Area [kPa]
100 80
C32 C33
60
C37 C38 C39
40
Table 4 Compressive failure properties.
20 0
0
1
2
3
4
5
6
7
8
9
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Displacement [mm] Fig. 13. Displacement-force per contact area progression of the tested specimens for which the failure pattern is displayed in Fig. 15.
Test no.
Crack origin offset from center
Load/contact area [kPa]
Load [N]
C32 C33 C37 C38 C39
1/2 of 1/2 of 1/3 of 3/4 of 0
76 103 74 103 70
51.6 73.0 48.0 72.2 45.9
half half half half
width width width width
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Measurement C35 Linear elastic simulation E=148MPa
70
Force [N]
60 50 40 30 20 10 0 −10
0
1
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4
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7
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Displacement [mm] Fig. 17. Comparison of a compressive specimen test to a linear elastic simulation run.
Fig. 15. Principal failure patterns from video analysis.
model-scale ice properties of the Aalto ice tank are not subjected to strain rate effects. In the compressive strength measurements it is found that the overall material behavior is unique, as all loading curves have a similar slope. However, the failure pattern varies with the point of failure. Li and Riska (2002) indicated that the voids in the model-scale ice govern the mechanical properties and the void distribution in the specimen may be responsible for the failure pattern. Due to the low load levels in model-scale, only micro-cracking is considered occurring as damage and neither pressure melting nor recrystallization is observed. The cause for the varying failure loads of the specimen is associated with the natural material inhomogeneity. The same reasoning is found for the variation of the tensile strength and its failure pattern. Kim and Sunder (1997) have been able to link the grain size distribution of fresh water ice with the distribution of the crack length. The data-set obtained from the strength-measurements in this paper is rather small. Hence, it is not possible yet to fit representative probability distributions to the strength-measurements to investigate their link to the micro-structure (grain size distribution, Fig. 8). Such investigation requires a larger data-set, however, the data-collection is practically restricted by the temperature sensitivity of the material. The temperature affects the mechanical properties, which are adjusted by the cooling history (see Fig. 1). The determined mechanical properties such as yield strength, elastic strain-modulus, tensile
strength and compressive strength are not universally valid, but only for the thickness and bending strength combination of this particular ice sheet. More parameters will be determined in the future following the procedure detailed in this paper. Fig. 18 displays the measured strength values over the time of measurement conduction. The time is relative to the first conducted measurement. According to Fig. 18, it is considered that the measured mechanical ice properties are not affected by transient changes. This means the additional cooling in the keeping phase (see Fig. 1) compensated the heat radiation of the model-scale ice well enough to maintain the mechanical ice properties. It must be noted that this does not always work, and hence the ice properties are often subjected to transient changes. Fig. 19 reflects data from similar ice property measurements in 25 mm thickness at the Aalto University basin with different testing speeds (see Wegner, 2011). Fig. 19 shows that in the tested speed ranges in which the linear drive can operate, the ice properties are not dependent on the testing speed and are considered strain rate insensitive. The testing speed of 6 mm/s used in this paper is chosen as it is the available medium speed level. 6. Conclusion Model-scale ice is commonly used for scaled experiments, but in many aspects the behavior of model-scale ice and its failure process are not yet explored. This paper indicates a method of determining the tensile strength, which is a parameter that is not well explored. Furthermore, the set of systematically conducted mode scale ice experiments allows the derivation of the basic material behavior, which is essential to build a numerical model that can solve engineering problems. The linear-elastic numerical analysis of the plate deflection experiment showed that the occurring stresses in thickness direction are large and their contribution to the deflection is significant. The elastic strain-modulus is found to be 26% larger when accounting for stresses in thickness direction than when following 120
Strength [kPa]
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Time [h] Fig. 16. Tensile failure of a tensile specimen.
Fig. 18. Measured strength values over the relative measurement time.
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Compressive Strength Tensile Strength
Strength [kPa]
100
81
Acknowledgments The authors would like to thank the technical personnel of the Aalto ice tank, J. Alasoini and T. Päivärinta for their practical support during the experiments and H. Ploskonka for the language check.
80 60
References
40 20 0
0
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4
6
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14
Testing speed [mm/s] Fig. 19. The impact of the testing speed on the strength measurement in a 25 mm thick ice sheet. Data from Wegner (2011).
plain stress theory. Furthermore, it is found that the elastic limit of the model-scale is very low and that a linear elastic model is not capable of reproducing the specimen tests. Consequently, this paper contributes to in-situ experiments for model-scale ice and their analysis to understand basic material behavior. The statistical analysis highlights the impact of the micro structure on the material response. Furthermore, the numerical analysis of the elastic strainmodulus and elastic behavior reveals that current state of the art analysis procedures may have deficiencies and that current assumptions regarding the material behavior are not valid. The gained knowledge indicates that the mechanics of model-scale ice are different from sea ice and provides the groundwork for building a numerical model that can be checked against the recorded time histories.
Derradji-Aouat, A., 1992. Mathematical Modelling of Monotonic and Cyclic Behaviour of Polycrystalline Fresh Water Ice. (PhD thesis) Carleton University. Elvin, A.A., Sunder, S.S., 1996. Microcracking due to grain boundary sliding in polycrystalline ice under uniaxial compression. Acta Materialia 44 (1), 43–56. Gammon, P., Kiefte, H., Clouter, M., Denner, W.W., 1983. Elastic constants of artificial and natural ice samples by Brillouin Spectroscopy. Journal of Glaciology 29 (29), 433–460. ITTC, 2002. Ice property measurements. http://ittc.sname.org. Jalonen, R., Ilves, L., 1990. Experience with a chemically-doped fine-grained model ice. IAHR Ice Symposium 1990 (2), 639–651. Kerr, A., Palmer, W., 1972. The deformations and stresses in floating ice plates. Acta Mechanica 15, 57–72. Kim, J., Sunder, S.S., 1997. Statistical effects on the evolution of compliance and compressive fracture stress of ice. Cold Regions Science and Technology 26 (2), 137–152. Li, Z., Riska, K., 1996. Preliminary study of physical and mechanical properties of model ice. Technical Report M-212. Helsinki University of Technology. Li, Z., Riska, K., 2002. Index for estimating physical and mechanical parameters of model ice. Journal of Cold Regions Engineering 16 (2), 72–82. Schwarz, J., 1985. Physical modeling techniques for offshore structures in ice. The 8th International Conference on Port and Ocean Engineering under ARCTIC Conditions, POAC 85, 3, pp. 1113–1131. Svec, O.J., Frederking, R.M.W., 1981. Cantilever beam tests in an ice cover: influence of plate effects at the root. Cold Regions Science and Technology 4 (2), 93–101. Timco, G., 1979. The mechanical and morphological properties of doped ice: a search for a better structurally simulated ice for model ice basins. Proceeding of the Port and Ocean Engineering under Arctic Conditions (POAC 79), 1, pp. 719–739. Timco, G.W., 1980. The mechanical properties of saline-doped and carbamide (urea)-doped model ice. Cold Regions Science and Technology 45–56. Timco, G.W., 1981. On the test methods for model ice. Cold Regions Science and Technology 4, 269–274. Timco, G., Weeks, W., 2010. A review of the engineering properties of sea ice. Cold Regions Science and Technology 60 (2), 107–129. TraFi, S.T.A., 2011. Guidelines for the Application the Finnish-Swedish Ice Class Rules. Wegner, K., 2011. Model Ice Property Measurements for FE-simulations. Master's thesis Technical University Hamburg-Harburg.
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Model-scale ice — Part A: Experiments Rüdiger von Bock und Polach a,b,⁎, Sören Ehlers b, Pentti Kujala a a b
Aalto University, School of Science and Technology, Department of Applied Mechanics, Marine Technology Group, P.O. Box 15300, FI-00076 Aalto, Finland Norwegian University of Science and Technology, Faculty of Engineering Science and Technology, Department of Marine Technology, NO-7491 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 11 October 2012 Accepted 3 July 2013 Keywords: Model-scale ice Experiments Strength measurements Thin plate theory Poisson's ratio Elastic strain-modulus
a b s t r a c t This paper is presenting novel model-scale ice property measurements for grain size, elastic strain-modulus, compressive and tensile specimen tests. The testing and analyzing procedure is targeted to define the basic material behavior accurately to understand the material behavior for the future development of a numerical material model. Additionally, the model-scale ice thickness and the bending strength (following ITTC) are determined to classify the ice properties. The experiments consist of systematic in-situ tests to identify the model-scale ice properties in a format suitable for numerical simulations. The elastic strain-modulus is determined on the intact level ice sheet based on the load displacement relationship of the infinite plate deflection. All specimens are cut with a template to minimize dimensional variations. The specimens are loaded with a linear drive at constant speed while displacement and force are recorded. The resulting load–displacement curves indicate good repeatability. The experiments are conducted over a time of 4 h–5 h in the keeping phase, where the cooling system is adjusted to maintain the mechanical ice properties, and the obtained results do not show a dependency on the time of testing. A linear-elastic finite element model is used to reproduce the plate bending measurements for the elastic strain-modulus determination. Therewith, it is found that the actual elastic strain-modulus is 27% larger than in plain stress theory due to stresses in thickness direction. Additionally, the approximate yield strength of the model-scale ice is investigated and is found to be significantly lower than the determined maximum stresses in compression, tension and bending. Consequently, this paper contributes to a deeper understanding of the mechanics of model-scale ice, and a procedure is shown how the mechanical parameters can be determined by systematic experiments and analyses. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Offshore operations and marine-borne transport systems are reaching more and more towards northern territories to exploit fossil resources and to ship goods through shorter, but ice infested, routes. The navigation in such areas requires the ships to be ice-capable. The Finnish-Swedish Ice-Class Rules TraFi (2011) have set the standard for the lower Polar ice/Baltic ice classes to which most classification societies refer. Those standards state the possibility to prove the ice capability of a ship with ice model tests. Additionally, the performance of most ice going structures and ships is tested in modelscale ice before concluding the design. Hence, the model-scale ice properties are of high significance for the design of marine structures, and a deeper understanding of model-scale ice as a material is important. In relation to full scale, most ice property measurements in model-scale ice can be conducted at low cost. Therefore, the model-scale ice offers the chance to investigate failure ⁎ Corresponding author at: Aalto University, School of Science and Technology, Department of Applied Mechanics, Marine Technology Group, P.O. Box 15300, FI-00076 Aalto, Finland. Tel.: +358 504 059 030. E-mail address: ruediger.vonbock@aalto.fi (R. von Bock und Polach). 0165-232X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coldregions.2013.07.001
processes and material behaviors better than in natural ice. This emphasizes conducting systematic ice property measurements to develop a numerical material based on the knowledge gained from the experiments. The numerical model of the model-scale ice targets future comprehension of marine design processes and simulations. Model-scale ice is grown in a controlled environment and the generated ice has a high potential of repeatability. The water from which the model-scale ice is created contains additives, such as ethanol at the Aalto University Ice Tank that makes the model-scale ice Froudescalable. The ice strength is affected by the properties of the grain boundaries. Elvin and Sunder (1996) described grain boundaries as the weakest link in polycrystalline ice due to their high entropy and disorder. The nature of the grain boundaries depends on the water dopant and the amount dissolved in water, which is reflected in Timco (1980). Therein, an extensive series of mechanical ice property measurements has been presented to explore new dopants to enhance the scalability of the model-scale ice properties. So far, model-scale ice property measurements are not reported in a systematic fashion suitable for a numerical implementation and are in general very scarce. Similarly, the results presented by Timco (1980) are not suitable for the implementation into numerical simulations for in-situ experiments, because the constitution of the ex-situ samples might have changed compared
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2 0
Temperature [C]
to in-situ conditions due to fluid drains (see Li and Riska, 1996; Timco, 1981). Furthermore, it was stated in Timco (1980) that the tensile strength could not be measured directly due to the weak constitution of the model-scale ice. Additionally, Timco (1980) used an empirical concentration factor to calculate the tensile strength, the origin of which might add an uncertainty to the results. Unfortunately, the tensile strength of ice and scaled ice (model-scale ice) is one of the least explored properties due to the difficulty involved in the measurements (see Timco and Weeks, 2010). Furthermore, the experiments of Timco (1980) are missing information on the entire material response up to specimen failure, which is essential for modeling the material behavior. In conclusion, earlier model-scale ice experiments are not suitable for in-situ simulations due to the lack of systematic measurements and the absence of a numerical implementation focus. The controlled model-scale ice-generation process allows adjusting target properties, but nevertheless (see Timco, 1980) the model testing results are subjected to some scatter. Earlier non-scaled ice experiments by Kim and Sunder (1997) found that both the obtained failure strength and the grain size distribution follow a log-normal probability distribution and established a connection between grain structure and failure. This indicates that the grain size distribution is dominating the strength compared to other micro-structural effects. This paper presents systematic ice property measurements and their analysis. The main objective of this paper is to gain a basic understanding of mechanical material behavior to provide sufficient information to develop a numerical material model in a later stage. The later numerical model aims to simulate ice–structure interactions and therefore all experiments are conducted in-situ i.e. the same environment where model-scale experiments are conducted. Model-scale ice is a unique and temperature (time) sensitive material, and constitutional changes between the tested samples are to be prevented. This naturally limits the number of samples in the analysis. Furthermore, all test specimens are of the same size to exclude variations due to geometrical effects. Finally, the grain size measurements are statistically analyzed to increase knowledge of the micro-structure and to investigate their link to the strength measurements in future research.
75
2
1
4
3
−2 −4 −6 −8 −10 −12 −14 0
5
10
15
20
25
30
Time [h] Fig. 1. Temperature time-history, with the phases of 1: spraying (adjustment of the thickness), 2: consolidation (adjustment of the strength), 3: tempering, and 4: keeping.
ice sheets), while the densities of the model-scale ice (911 kg/m3) and the basin water (989 kg/m3) were determined a priori. The density is determined from the buoyancy related reaction force of a submerged ice piece. 3.1. Grain size measurements The grain size measurements are conducted with post-processed photographs visualizing single grains with well-defined grain edges. The photographs are taken on a portable photographer-light-table, where a thin ice sample is placed between two transparent polar filter foils. As the material is very weak and unstable, thin sections are
a
35 30 25
2. The model-scale ice
20
x [m]
The model-scale ice is produced in the ice tank of the Aalto University, which is a square 40 m × 40 m basin with large cooling units over the entire basin widths on two opposite sides controlling the temperature. The model-scale ice is fine grained and is produced by the spraying method. The resulting model ice has fairly homogeneous structure of several crystal layers. More information is found in Jalonen and Ilves (1990), Li and Riska (1996) and Li and Riska (2002). Fig. 1 shows the cooling history and the different phases of cooling. The ice model tests are conducted in the keeping phase (phase 4, see Fig. 1), where sufficient cold air is added to keep the mechanical model-scale ice properties constant. Even though the ice properties are rather constant over the entire ice sheet, it is uncertain as to what extent the temperaturedistribution is a function of the tank location, i.e. the distance to the cooling units. Therefore, the experiments are only conducted in one particular section of the ice sheet, as seen in Fig. 2, where the tripod is covering a diameter of 4 m for size comparison.
15 10 5 0 −5 35
b
30
25
20
15
10
5
0
−5
y [m]
3. Experiments The measured ice properties in the ice sheet are the elastic strain-modulus, E, the bending strength, σb, the tensile strength, σt, and the compressive strength, σc. The latter two strength definitions refer to force divided by cross-sectional area. It is ambiguous whether the strength definitions are correct, but they refer to the state of the art in ice model testing tanks. The grain size is determined in later measurements in an ice sheet of 25 mm (the same thickness as the tested
Fig. 2. Location of experiments in the Aalto ice tank as (a) plan view and (b) testing site.
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Fig. 5. Template for specimen shapes for tension (1), compression (2) and bending (3). The loading direction of each specimen is indicated by the corresponding symbol.
to convert the pixels to millimeters. The length measurement in one direction suffices because the grains are fairly circular in this 2D view. Furthermore, it is assumed that this sample preparation (Fig. 3) is creating randomly orientated crystals, and thus the samples are assumed to represent the grain size in general.
3.2. Elastic strain-modulus
Fig. 3. Photographs of the three used grain size samples (a), (b) and (c).
impractical. The thin slice collapses during the cutting off and therefore the structure of the cross section cannot be analyzed. It is assumed that the collapsing of the slice leads to a random orientation of the grains on the polar filter foil. Subsequently, the polar filter foils are pressed slightly together by hand, without damaging the grains, so that the ice distributes as one spread layer of grains between the two foils. The foils are rotated relative to each other until the highest contrast with the ice is established, so that the single ice grains are clearly visible (see Fig. 3). The colors of the photographs are inverted to raise the clarity of the grain edges (see Fig. 3) and the grain size measurements are performed on three different photographs with different orientations with 20 sample measurements each. A millimeter scale is placed on each photograph for reference
Fig. 4. Measurement setup with the model-scale ice (1), the insulation plate (2), the steel mass (3), the emitted laser beam (4) and the laser (5).
The elastic strain-modulus of the ice sheet is determined from the deflection of the intact ice sheet, which is assumed as an infinite plate on an elastic foundation (following ITTC, 2002). The non-destructive tests are carried out prior to the destructive tensile, compressive and bending experiments. The ice sheet deflection is measured at four spots within the area of the later strength measurements. The ice sheet is loaded with three different certified mass levels of 26.9 g, 55.9 g and 109 g. Each measurement set includes several loading and unloading cycles of one mass level. The laser (5) in Fig. 4 is attached to a light-wooden tripod with a span of 4 m. The laser emits a beam (4) to measure the displacement difference between the loaded and unloaded condition on a coated Divinycell PVC foam plate, which is placed on the ice surface. The plate (2) prevents heat transfer between the steel mass (3) and the model-scale ice (1). However, it can be observed that despite cooling all equipment down to low temperature, a thin wet layer is generated between the equipment contact points and the ice sheet. This melting process leads to a continuous
Fig. 6. Specimen cutting process with specimens for tensile (1), compressive (2) and bending (3) experiments. See Fig. 5 for the related template shapes.
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Table 1 Measured elastic strain-modulus, E, for the ice sheet following ITTC (2002). Location
Mass [g]
Mean E [MPa]
1 2 3 4
26.9 55.9 109.0 109.0
108.2 108.1 Creep Creep
the ice thickness of 25 mm. The dimensions of the cantilever beam follow the ITTC guidelines (see ITTC, 2002). 4. Analysis of the experiments 4.1. Grain size Fig. 7. Test setup of the apparatus and the specimens in-situ with impact cylinder (1), impact plate (2), plunger (3), load-cell (4), linear drive (5) and rotation axis (6).
caving-in (0.1 mm) of the tripod into the ice, which, however, converges to a steady state. Therefore, both the load level and the zero level run parallel in a non-linear progression.
The analysis of the grain size dimensions is done by measuring 20 grains of each sample-photograph (a, b and c, see Fig. 3) for the statistical analysis. The grains selected have clearly visible edges and are best suited to size analysis. The statistical analyses of the 60 independent grain diameter measurements follow a log-normal distribution, with the highest probability density at 0.68 mm (see Fig. 8) and arithmetic mean of 0.72 mm (standard deviation of 0.13 mm).
3.3. Ice strength measurements 4.2. Elastic strain-modulus and ice sheet deflection The experiments are conducted in-situ by cutting specimens with a template in the ice sheet (as can be seen in Fig. 5). The numbers denote specimens for 1 — tension, 2 — compression, and 3 — bending (following ITTC, 2002). The dimensions of the specimens in Fig. 5 are related to the ice thickness, h, as follows: • Tensile specimen outer ring diameter = 3 h, inner ring diameter = h, and thinnest width of dumbbell-part = h • Compressive specimen width = h and length = h • Cantilever beam width = 2.25 h and length = 6.5 h. The specimens are cut with a guided drill to ensure the specimens are cut as vertical projections of the template. After the outlines of the specimens are cut, the surrounding ice is carefully removed by hand (see Fig. 6). Once the specimens are ready the testing device is employed. The testing device is a linear drive (see Fig. 7) equipped with a rigid impact cylinder (1) to apply the tensile loading, a rigid impact plate (2) to apply the compressive loads, a load-cell (4) and a displacement transducer (5). Further, orthogonal to the plate (2) is a plunger (3) for beam bending tests. In the bending tests the apparatus is rotated around axis 6 (Fig. 7) by 90° to load the cantilever beam vertically. The dimensions of all specimens are a function of the ice thickness. The width of the rated breaking point of the tensile specimen and the width of the compressive specimen were equal to 0.995 0.99
Table 1 compiles the results of the elastic strain-modulus measurements, which are analyzed based on the theory of the infinite plate on an elastic foundation. Table 1 shows a good agreement between measurement 1 and 2. As mentioned earlier, the displacement signal is subjected to some drift due to the carving-in of the tripod into the model-scale ice surface. Fig. 9 shows the drift of the displacement signal and the interpolation curves for the loaded sections and the zero levels. As it is known that the signal drift (see also Li and Riska, 1996) would converge to a certain value after the process of caving-in is terminated, a cubic function is chosen instead of a linear interpolation. The time history sections under load should ideally be a curve with a constant offset to the unloaded curve sections. Significant deviations from this offset indicate creep, as reflected in Fig. 10, where the load of 109 g indicates exceedance of the linear-elastic regime. The numerical model is built in LS-Dyna, where the ice sheet is modeled as a homogeneous, isotropic and linear-elastic material, with the material properties elastic strain-modulus, E, and Poisson's ratio, v. The elastic foundation is modeled with load displacement nodes where a displacement of the nodes causes a reaction force equivalent to the buoyancy restoring force. The modeled ice sheet measures 4 m × 4 m with all degrees of freedom constrained at the edges and 25 mm sized elements. The load on the ice sheet is applied
0.1
Grain size measurements Log−normal (N = 60, p = 0.0026)
Displacement [mm]
0.95
Probability
0.9 0.75 0.5 0.25 0.1 0.05 0.01
Measurement Zero−level Load−level
0.05 0 −0.05 −0.1
0.005 0.001 0.5
0.6
0.7
0.8
Grain size [mm] Fig. 8. Measured grain sizes with log-normal probability plot.
0.9
1
0
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50
60
70
Time [s] Fig. 9. Displacement measurement of infinite plate bending with 26.9 g loading.
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Table 2 The stresses for the solid model at the maximum displacement at 55.9 g loading.
Displacement [mm]
0.15 0.1 0.05
Effective stress [Pa]
σx [Pa]
σy [Pa]
σz [Pa]
τxy [Pa]
τzy [Pa]
148
449
32
32
57
13.8
172
0 −0.05 −0.1 −0.15
Start of creeping
−0.2 −0.25 0
50
100
150
200
250
300
350
400
Time [s] Fig. 10. Displacement measurement of infinite plate bending with 109 g loading.
as a point load (see also ITTC, 2002). The ice sheet deflection is successfully modeled with shell elements complying with the analytical formulation for a thin plate on an elastic foundation stated in ITTC (2002). However, the remodeling of the plate deflection with fully integrated solids results in a higher deflection by 12.3% (see Fig. 11). Compliance between the displacement of the solid FE-model and the measurements is achieved by increasing the elastic moduli to 25%–27% (145 MPa and 148 MPa). Table 2 compiles the stresses for the 55.9 g mass unit and states that the stresses in throughthickness direction σz and the shear stresses τzy are of significant magnitudes, which are zero for the thin plate theory following plane-stress theory. The ice shows significant inelastic behavior at 109 g loading (see Fig. 10). The maximum effective elastic stress (von Mises) for elastic response is found for the 55.9 g mass unit at 449 Pa (see Table 2). The elastic limit of the material is of high significance in a numerical model. In the linear-elastic domain, the Poisson's ratio is the single parameter affected with uncertainty. Table 3 reflects the sensitivity of the elastic strain-modulus and the maximum effective stress (von Mises) in dependence on the Poisson's ratio at the same maximum displacement.
4.3. Strength measurements The average (downward) bending strength following ITTC (2002), without accounting for buoyancy effects, is found to be 59 kPa. Fig. 12 reflects the directly measured force displacement curve of the compression measurements. Figs. 13 and 14 show the force divided by the cross-sectional area of the specimen at the location of failure. The Figs. 12–14 show that the slopes of all specimens are rather similar. However, the point of failure is different, which is a phenomenon 0.01 0
Displacement [mm]
Elastic strain-modulus [MPa]
Solid model Shell model
−0.01 −0.02 −0.03 −0.04
that is also known from other materials such as concrete and even steel. The compressive and tensile failure processes are analyzed and interpreted in context with the visual observations, i.e. video recordings, and the recorded signals. In compression, the ice appears to undergo a linear deformation until failure. In some cases the slope of the curve reduces prior to failure, which indicates additional softening at a certain stress level. The videos revealed that the failure initiates in the center of the specimen width, where the specimen experiences high confinement. Therefrom, the failure propagates towards the high energetic sides (free surface energy). The crack initiation requires a nucleation base, which may be impurities or small defects in combination with high local confinement. Furthermore, Table 4 states the origin of the failure crack progression. Table 4 does not indicate a correlation between the post-compressive crack pattern and failure load level. The tensile failure occurs at the predetermined breaking point, where the specimen has the thinnest cross-section, as seen in Fig. 16. The crack occurs perpendicular to the direction of load application. Five of the specimens have a crack perpendicular to the load direction and one each with an angle of 74° and 68° to the direction of the load application. As for the compressive tests, no correlation between load level and breaking pattern is indicated. The maximum stresses in the tensile and compressive specimens are determined by assuming that the stresses are evenly distributed over the cross-section. Possible material inhomogeneities are neglected, which complies with current approaches in engineering.
5. Discussion The density of the model-scale ice is measured as 911 kg/m3. The model-scale ice is considered a compound of ice crystals, water voids and air voids. The density of frozen water (ice) is 917 kg/m3, the density of the tank water is 989 kg/m3 (measured) and the density of air is 1.3 kg/m3. Based on this, the volume fractions of the different compounds are determined as 4.5% water inclusions and 1% air inclusions, which is in line with the description of Li and Riska (2002). The grain size distribution is obtained from randomly selected samples of each test. Therefore, the selected grains reflect the global state of the model-scale ice texture. The log-normal distribution is found to fit best to the grain sizes. The grain size distribution of the model-scale ice coincides with the finding of Kim and Sunder (1997) for fresh-water ice, yet it is not clear whether this is coincidence or if there is a direct link. The elastic strain-modulus is determined by the deflection of an elastic plate on an elastic foundation. It is assumed that elastic strain-modulus is constant and isotropic over the thickness. The temperature gradient between top and bottom differs at maximum by 0.05 °C. Therefore, it is assumed that, as in nature, a temperature related variation over the ice thickness of the elastic strain-modulus
−0.05 Table 3 Elastic strain-modulus and effective stress in dependence on the Poisson's ratio.
−0.06 −0.07 0
500
1000
1500
Distance [mm] Fig. 11. The bending lines of plates on an elastic foundation modeled with shells and solids.
Poisson's ratio
Elastic strain-modulus [MPa]
Effective stress [Pa]
0.3 0.2 0.4
148 134 158
449 450.1 448
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40 30 20 10 0
T31
60
T32 T33
50
Force / Area [kPa]
Force [N]
50
−10
70
C32 C33 C37 C38 C39
60
79
T35
40
T36 T38
30
T39
20 10 0
−10 0
1
2
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4
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6
7
8
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Displacement [mm]
7
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Displacement [mm]
Fig. 12. Displacement-force progression of the tested specimens for which the failure pattern is displayed in Fig. 15.
Fig. 14. Displacement-force per contact area for the tensile measurements.
(see Kerr and Palmer, 1972) or of other mechanical parameters is negligible. The measured displacement signal from the elastic strain-modulus measurements is subjected to drift. The drift is related to a caving-in of the tripod feet into the ice, which converges to a steady state over time. Therefore, the zero and load level are interpolated with a cubic function. The Poisson's ratio, v, is the only parameter subjected to uncertainty, as it is very difficult to determine for model ice due to the weak constitution. Li and Riska (1996) and Timco (1979) (Aalto ice tank) used v = 0.3 for model ice, which is in line with similar values used for full scale scenarios by Gammon et al. (1983) and Svec and Frederking (1981) or more advance mathematical models such as Derradji-Aouat (1992). This value is also in line with the theory of Froude-scaling. Fig. 10 shows that the model-scale ice is subjected to creep/ permanent deformation for a mass of 109 g. The load level of 56 g is the highest test load for which the model-scale ice responded elastically. The corresponding stress state is considered to represent the lower confident yield limit of the model-scale ice. The remaining uncertainty in the Poisson's ratio affects the yield strength (Table 3) negligibly. The finite element analyses with solids and shells deliver a difference in the elastic strain-modulus by 26% for the same displacement. Fig. 11 reflects the significant difference in the bending lines of the solid and the shell model for the same elastic strain-modulus and the same load. Furthermore, the shell model is capable of remodeling the infinite plate on an elastic foundation exactly, following ITTC (2002). The difference in displacement is related to the relatively large stresses (shear and normal) in thickness direction, which are
not regarded in the thin plate theory following the plain stress approach. It must be acknowledged that the plate deflection experiment is a quasi-static test, whereas the compressive, tensile or bending tests are dynamic tests with a loading rate. However, the static test is considered sufficient, which is common practice in all model test basins. The determined yield strength (0. kPa) lies about two orders of magnitude below the value ranges of bending strength, compressive strength or tensile strength. This indicates that the linear elastic range is small and most of the deformation process takes place in a damage driven domain. Herein it is assumed that the yield surface is isotropic. Fig. 17 shows that the linear elastic simulation based on the measured elastic strain-modulus cannot represent the measured load displacement curve. This is in line with the finding of a relatively low yield strength, which indicates that the deformation process must take place in a domain other than the linear elastic one. The strong deviation of the linear elastic force displacement curve from the measured load displacement curve indicates that the occurring effects are not viscoelastic. The progressions in Fig. 17 suggest that after yielding the stress–strain behavior is governed by a lower tangent or softening modulus. Earlier research of Schwarz (1985) and Timco (1980) indicated the elastic-strain modulus as key parameter to be scaled in model-scale ice tests. It is considered to drive the deformation behavior. This complies with the idea of Froude-scaling and the understanding of sea-ice behavior (see Timco and Weeks, 2010), which is considered as mainly elastic. However, it must be acknowledged that research as presented in this paper has not been presented yet for any model-scale ice. In this context references of yield or creep strength of model-scale ice are not available. The constitution of model ice is slushier and differs significantly from sea ice, which is dominated by elastic and viscoelastic effects. This in turn challenges the postulation of Schwarz (1985) that the ratio of elastic strain-modulus and bending strength can be used as quality criterion for the scalability of the model-scale ice of Aalto University or model-scale ice in general. All experiments (tensile, compressive and bending) are conducted at a testing speed of 6 mm/s, as Wegner (2011) indicated that between 6 mm/s and 17 mm/s the
120
Force / Area [kPa]
100 80
C32 C33
60
C37 C38 C39
40
Table 4 Compressive failure properties.
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1
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Displacement [mm] Fig. 13. Displacement-force per contact area progression of the tested specimens for which the failure pattern is displayed in Fig. 15.
Test no.
Crack origin offset from center
Load/contact area [kPa]
Load [N]
C32 C33 C37 C38 C39
1/2 of 1/2 of 1/3 of 3/4 of 0
76 103 74 103 70
51.6 73.0 48.0 72.2 45.9
half half half half
width width width width
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Measurement C35 Linear elastic simulation E=148MPa
70
Force [N]
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4
5
6
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Displacement [mm] Fig. 17. Comparison of a compressive specimen test to a linear elastic simulation run.
Fig. 15. Principal failure patterns from video analysis.
model-scale ice properties of the Aalto ice tank are not subjected to strain rate effects. In the compressive strength measurements it is found that the overall material behavior is unique, as all loading curves have a similar slope. However, the failure pattern varies with the point of failure. Li and Riska (2002) indicated that the voids in the model-scale ice govern the mechanical properties and the void distribution in the specimen may be responsible for the failure pattern. Due to the low load levels in model-scale, only micro-cracking is considered occurring as damage and neither pressure melting nor recrystallization is observed. The cause for the varying failure loads of the specimen is associated with the natural material inhomogeneity. The same reasoning is found for the variation of the tensile strength and its failure pattern. Kim and Sunder (1997) have been able to link the grain size distribution of fresh water ice with the distribution of the crack length. The data-set obtained from the strength-measurements in this paper is rather small. Hence, it is not possible yet to fit representative probability distributions to the strength-measurements to investigate their link to the micro-structure (grain size distribution, Fig. 8). Such investigation requires a larger data-set, however, the data-collection is practically restricted by the temperature sensitivity of the material. The temperature affects the mechanical properties, which are adjusted by the cooling history (see Fig. 1). The determined mechanical properties such as yield strength, elastic strain-modulus, tensile
strength and compressive strength are not universally valid, but only for the thickness and bending strength combination of this particular ice sheet. More parameters will be determined in the future following the procedure detailed in this paper. Fig. 18 displays the measured strength values over the time of measurement conduction. The time is relative to the first conducted measurement. According to Fig. 18, it is considered that the measured mechanical ice properties are not affected by transient changes. This means the additional cooling in the keeping phase (see Fig. 1) compensated the heat radiation of the model-scale ice well enough to maintain the mechanical ice properties. It must be noted that this does not always work, and hence the ice properties are often subjected to transient changes. Fig. 19 reflects data from similar ice property measurements in 25 mm thickness at the Aalto University basin with different testing speeds (see Wegner, 2011). Fig. 19 shows that in the tested speed ranges in which the linear drive can operate, the ice properties are not dependent on the testing speed and are considered strain rate insensitive. The testing speed of 6 mm/s used in this paper is chosen as it is the available medium speed level. 6. Conclusion Model-scale ice is commonly used for scaled experiments, but in many aspects the behavior of model-scale ice and its failure process are not yet explored. This paper indicates a method of determining the tensile strength, which is a parameter that is not well explored. Furthermore, the set of systematically conducted mode scale ice experiments allows the derivation of the basic material behavior, which is essential to build a numerical model that can solve engineering problems. The linear-elastic numerical analysis of the plate deflection experiment showed that the occurring stresses in thickness direction are large and their contribution to the deflection is significant. The elastic strain-modulus is found to be 26% larger when accounting for stresses in thickness direction than when following 120
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Time [h] Fig. 16. Tensile failure of a tensile specimen.
Fig. 18. Measured strength values over the relative measurement time.
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Acknowledgments The authors would like to thank the technical personnel of the Aalto ice tank, J. Alasoini and T. Päivärinta for their practical support during the experiments and H. Ploskonka for the language check.
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References
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Testing speed [mm/s] Fig. 19. The impact of the testing speed on the strength measurement in a 25 mm thick ice sheet. Data from Wegner (2011).
plain stress theory. Furthermore, it is found that the elastic limit of the model-scale is very low and that a linear elastic model is not capable of reproducing the specimen tests. Consequently, this paper contributes to in-situ experiments for model-scale ice and their analysis to understand basic material behavior. The statistical analysis highlights the impact of the micro structure on the material response. Furthermore, the numerical analysis of the elastic strainmodulus and elastic behavior reveals that current state of the art analysis procedures may have deficiencies and that current assumptions regarding the material behavior are not valid. The gained knowledge indicates that the mechanics of model-scale ice are different from sea ice and provides the groundwork for building a numerical model that can be checked against the recorded time histories.
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